Let $C$ be a Frobenius category. The stable category $\underline{C}$ is called $m$-Calabi Yau in case it is Hom-finite and there is a functorial duality $D \underline{Hom}(X,Y)=\underline{Hom}(Y,\Omega^{-m}(X))$ for $X,Y \in C$.
Question: In case we have just $\dim( \underline{Hom}(X,Y))=\dim(\underline{Hom}(Y,\Omega^{-m}(X)))$ for $X,Y \in C$ (and Hom-finiteness) instead, is this enough for being $m$-Calabi Yau?
Im especially interested when $C$ is the module category of a Frobenius algebra.
